Definition of Concavity and Inflection Points

The terms concavity and inflection point refer to the directionality of a curve. If a curve is concave up (convex), the graph of the curve is bent upward, like an upright bowl. If a curve is concave down (or simply concave), then the graph of the curve is bent down, like a bridge. For a function f(x) where f(x) and f′(x) are both differentiable, f(x) is concave up if and concave down if . If , then x is an inflection point, where the graph changes direction of concavity. The inflection point theorem states that if exists and changes sign at , then is an inflection point. If exists, then .